Simplify the following expression and state the condition under which the simplification is valid: $z = \dfrac{k^2 + 9k}{k^2 + 10k + 9}$
First factor the expressions in the numerator and denominator. $ \dfrac{k^2 + 9k}{k^2 + 10k + 9} = \dfrac{(k)(k + 9)}{(k + 1)(k + 9)} $ Notice that the term $(k + 9)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(k + 9)$ gives: $z = \dfrac{k}{k + 1}$ Since we divided by $(k + 9)$, $k \neq -9$. $z = \dfrac{k}{k + 1}; \space k \neq -9$